Here is an instant-answer trick that works on a surprisingly large fraction of JEE arithmetic problems. When a question gives you percentage changes but no actual starting number — "a price is raised by 20\% and then lowered by 15\%, what is the net change?" — your first move should not be algebra. Your first move should be: plug in 100.

Set the unknown starting value to ₹100. Run the changes on it. Read the answer straight off the final number.

100 \xrightarrow{+20\%} 120 \xrightarrow{-15\%} 102

Final value ₹102, so the net change is +2\%. Done. No algebra, no variables, no (1+p/100) in the working. The reason 100 works is that percentages are by definition amounts per hundred, so 100 turns every percent into a rupee and every multiplication into a clean decimal.

Why 100 is the magic number

Percentages are fractions with denominator 100 — the Latin per centum literally means "per hundred." When the starting quantity is already 100, every percentage change lands on a nice integer without any scaling. A 23\% rise adds 23; a 7\% fall subtracts 7. The arithmetic runs on its own.

Why the answer doesn't depend on the starting value: percentage changes are proportional operations — they scale linearly with the base. If a chain of percentage changes turns 100 into 102, the same chain turns X into 1.02\, X for every X. So whatever "net percent" you read off when you start from 100 is the net percent for the original problem, guaranteed. Starting from 100 just removes the messy fractions.

Starting from 1000 or 10{,}000 works too, but 100 gives the smallest sensible integers. Starting from 1 gives decimals. 100 is the sweet spot.

Plug-in-one-hundred slider for a price raised by twenty percent then lowered by fifteen percentThree horizontal bars stacked vertically. The top bar is always labelled one hundred rupees, the starting value. The middle bar grows to one hundred twenty rupees after a plus twenty percent markup. The bottom bar shrinks to one hundred two rupees after a fifteen percent discount. A slider lets you change the second discount from zero percent down to minus fifty percent and back up, and the three bars resize so you can watch how the final rupee number tracks the net percentage change. ₹100 starting ₹120 after +20% −50% 0% +50% ↔ drag to change second percent
Start from ₹$100$ (top bar). A $+20\%$ markup gives ₹$120$ (middle). Then drag the red dot to pick the second percentage change. At $-15\%$ the final bar reads ₹$102$ — a $2\%$ net gain. At $-20\%$ it reads ₹$96$, a $4\%$ net loss. The number on the bottom bar *is* the net percent change, because you started from $100$.

Four problems, no algebra needed

Problem 1. A salary is increased by 10\%, then by another 20\%. What is the overall percentage increase?

Start at 100. After +10\%: 110. After +20\%: 110 \times 1.2 = 132. Overall increase: 32\%.

Problem 2. A shop marks an item up 50\% then offers a 30\% discount. Profit or loss, and by how much?

Start at 100. After +50\%: 150. After -30\%: 150 \times 0.7 = 105. Net profit: 5\%.

Problem 3. The length of a rectangle is increased by 20\% and the breadth decreased by 10\%. What is the percentage change in area?

Start both at 100, so area is 10{,}000. New length: 120. New breadth: 90. New area: 120 \times 90 = 10{,}800. Change: +8\%.

Problem 4. A's income is 25\% more than B's. B's income is what percentage less than A's?

Let B's income be 100. Then A's is 125. B is less than A by 25, and as a percent of A that is \frac{25}{125} \times 100 = 20\%. (The asymmetry here — 25\% up vs 20\% down — is the classic asymmetry that catches students. Plug-in-100 makes it painless.)

All four problems solved in four short lines each, with no variables introduced.

When to use the trick

The plug-in-100 habit is at its best when:

It is less useful when the problem already gives you the starting number (just use it), or when the answer must be an absolute quantity like "how many rupees?".

The natural extension — plug in what the problem wants

The 100 trick is a special case of a bigger idea: when a problem has a free parameter, plug in a convenient value. For area problems, plug in 10 for both sides. For ratio problems, plug in the smallest integers that fit the ratio. For time-and-work problems, plug in the LCM of the times. The principle is always the same: solve the concrete version of the problem, trust that the symbolic version has the same answer.

The two rules of plug-in

Rule 1. You can only plug in for a quantity the problem treats as a free variable. If the problem says "the price is ₹80," you must use 80, not 100.

Rule 2. Plug in the same value everywhere that variable appears. If A is "25\% more than B" and you pick B = 100, then A = 125 everywhere in the problem — not 100 in one place and 125 in another.

Break either rule and the answer will be wrong — not because plug-in is broken, but because you changed the problem.

Reflex checklist

When you read a percentage problem and notice no concrete number has been given:

  1. Set the unknown base to 100 (or, for ratios, the smallest integers that fit).
  2. Walk through each percentage change, computing the new number each time.
  3. Compare the final number to 100 to read off the net percent change.
  4. If the answer needs to be an absolute number (rupees, kilograms), re-insert the real starting value at the end.

This is one of the cheapest, highest-leverage habits in JEE arithmetic. Once you have trained your hands to write "100" the instant you see a percentage problem with no number, you will solve chains of percentage changes faster than you can set up the equation.

Related: Percentages and Ratios · Every Percentage Change Is a Multiplier — +25% is 1.25, −25% is 0.75 · See Successive Percentage Changes → Multiply Factors, Never Add Them · How Do You Find the Original Price When Only the Discounted Price and Discount % Are Given?